The ability to group message with others relies on MMS rather than SMS. By using MMS, phones involved in the conversation are able to keep track not just of the original sender but also about everyone else involved. This is true for Windows Phone and I am quite sure for all other platforms as well.
Like you mentioned in your question, when you have a group ...
Do you have your Facebook account linked "properly" to your windows phone account?
You can still chat without it, but I wasn't receiving all messages, I'd be receiving some, I'd reply, but I wouldn't receive the message in reply to mine.
Once I linked them, it was working. To check if you have linked them properly:
Log into outlook.com
Go to Account ...
The first tick indicates that the message has been delivered to the Facebook server.
The second tick indicates that the message has been delivered to the recipient's phone. It doesn't necessarily mean that the recipient has read it.
If the profile picture of the recipient appears, instead of the tick it means that the recipient has read your message.
AFAIK the best option would be Threema. This messaging tool is made by a swiss company supporting end-to-end encryption and no need for any personal data if you don't want. According to the company any data will be deleted from their servers as soon as a message was delivered. I haven't tried it myself yet but I heard some good things.
Looks like that version of Messenger is no longer supported on Windows Phone 8.1. Here's an official announcement from Facebook stating that various apps would be discontinued on older operating systems, including WP8 as well as older versions of iOS and Android:
Update on messaging support for older versions of Messenger and Facebook apps
Yes it is true that you can use messenger app without a facebook account but the WP app does not support that functionality. You may consider creating a facebook account. That's not a very big deal. Otherwise you can't use messenger
Facebook has developed its applications i.e. Facebook and Messenger differently for Windows Phone 8.x and Windows 10 Mobile. So you don't have to be worried about restrictions of W10M variant of Messenger while installing Messenger on a 8.1 device. I don't know whether it supports 520 or not as I never dealt with <1GB RAM devices but if 8.x variant does ...